aboutsummaryrefslogtreecommitdiff
path: root/docs/articles
diff options
context:
space:
mode:
authorJohannes Ranke <jranke@uni-bremen.de>2017-03-06 17:00:48 +0100
committerJohannes Ranke <jranke@uni-bremen.de>2017-03-06 17:00:48 +0100
commit73e650114af77582238abf5273e63005e0b2287e (patch)
treee0effa0124a713738c850aa47acc54d1b82318d8 /docs/articles
parentfcb3332b7b186639cce940696057a2b6dd2c5cca (diff)
Static documentation now built by pkgdown::build_site()
Diffstat (limited to 'docs/articles')
-rw-r--r--docs/articles/chemCal.Rnw130
-rw-r--r--docs/articles/index.html98
2 files changed, 228 insertions, 0 deletions
diff --git a/docs/articles/chemCal.Rnw b/docs/articles/chemCal.Rnw
new file mode 100644
index 0000000..30e0331
--- /dev/null
+++ b/docs/articles/chemCal.Rnw
@@ -0,0 +1,130 @@
+\documentclass[a4paper]{article}
+%\VignetteIndexEntry{Short manual for the chemCal package}
+\usepackage{hyperref}
+
+\title{Basic calibration functions for analytical chemistry}
+\author{Johannes Ranke}
+
+\begin{document}
+\maketitle
+
+The \texttt{chemCal} package was first designed in the course of a lecture and lab
+course on "analytics of organic trace contaminants" at the University of Bremen
+from October to December 2004. In the fall 2005, an email exchange with
+Ron Wehrens led to the belief that it would be desirable to implement the
+inverse prediction method given in \cite{massart97} since it also covers the
+case of weighted regression. Studies of the IUPAC orange book and of DIN 32645
+as well as publications by Currie and the Analytical Method Committee of the
+Royal Society of Chemistry and a nice paper by Castillo and Castells provided
+further understanding of the matter.
+
+At the moment, the package consists of four functions, working on univariate
+linear models of class \texttt{lm} or \texttt{rlm}, plus to datasets for
+validation.
+
+A \href{http://bugs.r-project.org/bugzilla3/show_bug.cgi?id=8877}{bug
+report (PR\#8877)} and the following e-mail exchange on the r-devel mailing list about
+prediction intervals from weighted regression entailed some further studies
+on this subject. However, I did not encounter any proof or explanation of the
+formula cited below yet, so I can't really confirm that Massart's method is correct.
+
+When calibrating an analytical method, the first task is to generate a suitable
+model. If we want to use the \texttt{chemCal} functions, we will have to restrict
+ourselves to univariate, possibly weighted, linear regression so far.
+
+Once such a model has been created, the calibration can be graphically
+shown by using the \texttt{calplot} function:
+
+<<echo=TRUE,fig=TRUE>>=
+library(chemCal)
+data(massart97ex3)
+m0 <- lm(y ~ x, data = massart97ex3)
+calplot(m0)
+@
+
+As we can see, the scatter increases with increasing x. This is also
+illustrated by one of the diagnostic plots for linear models
+provided by R:
+
+<<echo=TRUE,fig=TRUE>>=
+plot(m0,which=3)
+@
+
+Therefore, in Example 8 in \cite{massart97} weighted regression
+is proposed which can be reproduced by
+
+<<>>=
+attach(massart97ex3)
+yx <- split(y, x)
+ybar <- sapply(yx, mean)
+s <- round(sapply(yx, sd), digits = 2)
+w <- round(1 / (s^2), digits = 3)
+weights <- w[factor(x)]
+m <- lm(y ~ x, w = weights)
+@
+
+If we now want to predict a new x value from measured y values,
+we use the \texttt{inverse.predict} function:
+
+<<>>=
+inverse.predict(m, 15, ws=1.67)
+inverse.predict(m, 90, ws = 0.145)
+@
+
+The weight \texttt{ws} assigned to the measured y value has to be
+given by the user in the case of weighted regression, or alternatively,
+the approximate variance \texttt{var.s} at this location.
+
+\section*{Theory for \texttt{inverse.predict}}
+Equation 8.28 in \cite{massart97} gives a general equation for predicting the
+standard error $s_{\hat{x_s}}$ for an x value predicted from measurements of y
+according to the linear calibration function $ y = b_0 + b_1 \cdot x$:
+
+\begin{equation}
+s_{\hat{x_s}} = \frac{s_e}{b_1} \sqrt{\frac{1}{w_s m} + \frac{1}{\sum{w_i}} +
+ \frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
+ {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} -
+ {\left( \sum{ w_i x_i } \right)}^2 \right) }}
+\end{equation}
+
+with
+
+\begin{equation}
+s_e = \sqrt{ \frac{\sum w_i (y_i - \hat{y_i})^2}{n - 2}}
+\end{equation}
+
+where $w_i$ is the weight for calibration standard $i$, $y_i$ is the mean $y$
+value (!) observed for standard $i$, $\hat{y_i}$ is the estimated value for
+standard $i$, $n$ is the number calibration standards, $w_s$ is the weight
+attributed to the sample $s$, $m$ is the number of replicate measurements of
+sample $s$, $\bar{y_s}$ is the mean response for the sample,
+$\bar{y_w} = \frac{\sum{w_i y_i}}{\sum{w_i}}$ is the weighted mean of responses
+$y_i$, and $x_i$ is the given $x$ value for standard $i$.
+
+The weight $w_s$ for the sample should be estimated or calculated in accordance
+to the weights used in the linear regression.
+
+I adjusted the above equation in order to be able to take a different
+precisions in standards and samples into account. In analogy to Equation 8.26
+from \cite{massart97} we get
+
+\begin{equation}
+s_{\hat{x_s}} = \frac{1}{b_1} \sqrt{\frac{{s_s}^2}{w_s m} +
+ {s_e}^2 \left( \frac{1}{\sum{w_i}} +
+ \frac{(\bar{y_s} - \bar{y_w})^2 \sum{w_i}}
+ {{b_1}^2 \left( \sum{w_i} \sum{w_i {x_i}^2} - {\left( \sum{ w_i x_i } \right)}^2 \right) } \right) }
+\end{equation}
+
+where I interpret $\frac{{s_s}^2}{w_s}$ as an estimator of the variance at location
+$\hat{x_s}$, which can be replaced by a user-specified value using the argument
+\texttt{var.s} of the function \texttt{inverse.predict}.
+
+\begin{thebibliography}{1}
+\bibitem{massart97}
+Massart, L.M, Vandenginste, B.G.M., Buydens, L.M.C., De Jong, S., Lewi, P.J.,
+Smeyers-Verbeke, J.
+\newblock Handbook of Chemometrics and Qualimetrics: Part A,
+\newblock Elsevier, Amsterdam, 1997
+\end{thebibliography}
+
+\end{document}
diff --git a/docs/articles/index.html b/docs/articles/index.html
new file mode 100644
index 0000000..5c6fdad
--- /dev/null
+++ b/docs/articles/index.html
@@ -0,0 +1,98 @@
+<!-- Generated by pkgdown: do not edit by hand -->
+<!DOCTYPE html>
+<html>
+ <head>
+ <meta charset="utf-8">
+<meta http-equiv="X-UA-Compatible" content="IE=edge">
+<meta name="viewport" content="width=device-width, initial-scale=1.0">
+
+<title>Articles • chemCal</title>
+
+<!-- jquery -->
+<script src="https://code.jquery.com/jquery-3.1.0.min.js" integrity="sha384-nrOSfDHtoPMzJHjVTdCopGqIqeYETSXhZDFyniQ8ZHcVy08QesyHcnOUpMpqnmWq" crossorigin="anonymous"></script>
+<!-- Bootstrap -->
+
+<link href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css" rel="stylesheet" integrity="sha384-BVYiiSIFeK1dGmJRAkycuHAHRg32OmUcww7on3RYdg4Va+PmSTsz/K68vbdEjh4u" crossorigin="anonymous">
+<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js" integrity="sha384-Tc5IQib027qvyjSMfHjOMaLkfuWVxZxUPnCJA7l2mCWNIpG9mGCD8wGNIcPD7Txa" crossorigin="anonymous"></script>
+
+<!-- Font Awesome icons -->
+<link href="https://maxcdn.bootstrapcdn.com/font-awesome/4.6.3/css/font-awesome.min.css" rel="stylesheet" integrity="sha384-T8Gy5hrqNKT+hzMclPo118YTQO6cYprQmhrYwIiQ/3axmI1hQomh7Ud2hPOy8SP1" crossorigin="anonymous">
+
+
+<!-- pkgdown -->
+<link href="../pkgdown.css" rel="stylesheet">
+<script src="../jquery.sticky-kit.min.js"></script>
+<script src="../pkgdown.js"></script>
+
+<!-- mathjax -->
+<script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'></script>
+
+<!--[if lt IE 9]>
+<script src="https://oss.maxcdn.com/html5shiv/3.7.3/html5shiv.min.js"></script>
+<script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script>
+<![endif]-->
+ </head>
+
+ <body>
+ <div class="container template-vignette-index">
+ <header>
+ <div class="navbar navbar-default navbar-fixed-top" role="navigation">
+ <div class="container">
+ <div class="navbar-header">
+ <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar">
+ <span class="icon-bar"></span>
+ <span class="icon-bar"></span>
+ <span class="icon-bar"></span>
+ </button>
+ <a class="navbar-brand" href="../index.html">chemCal</a>
+ </div>
+ <div id="navbar" class="navbar-collapse collapse">
+ <ul class="nav navbar-nav">
+ <li>
+ <a href="../reference/index.html">Reference</a>
+</li>
+<li>
+ <a href="../articles/index.html">Articles</a>
+</li>
+ </ul>
+
+ <ul class="nav navbar-nav navbar-right">
+
+ </ul>
+ </div><!--/.nav-collapse -->
+ </div><!--/.container -->
+</div><!--/.navbar -->
+
+
+ </header>
+
+ <div class="page-header">
+ <h1>Vignette reference <small>version&nbsp;0.1-37.9000</small></h1>
+</div>
+
+<div class="row">
+ <div class="col-md-9">
+ <div class="section ">
+ <h3>All vignettes</h3>
+ <p class="section-desc"></p>
+
+ <ul>
+ </ul>
+ </div>
+ </div>
+</div>
+
+ <footer>
+ <div class="copyright">
+ <p>Developed by Johannes Ranke.</p>
+</div>
+
+<div class="pkgdown">
+ <p>Site built with <a href="http://hadley.github.io/pkgdown/">pkgdown</a>.</p>
+</div>
+
+ </footer>
+ </div>
+
+ </body>
+</html>

Contact - Imprint